Digital Simulation in Electrochemistry (eBook)

Dieter Britz, Ph.D. (Sydney Univ. 1967), Dipl. Comp. Sci. (University of Newcastle, Australia, 1985), Dr. scient (Aarhus Univ., Denmark, 2007). Dr. Britz has gathered longstanding experience in electrochemistry during research stays all over the world: he worked at the CSIRO, Sydney, on corrosion problems, on inorganic ion exchangers at the University of New York at Buffalo (1967-68), he performed instrumental work at the University of Kentucky, Lexington, USA (1968-70), investigated corrosion and electrosynthesis at the Nuclear Research Centre in Jülich, Germany (1970-75), and performed data analysis of turbulence signals at Newcastle University, Australia (1975-78). In 1978 he accepted the position of Assoc. Professor at Aarhus University in Denmark, from which he retired as Emeritus Assoc. Professor in 2001. In Aarhus, he has worked on a number of projects, focusing on corrosion, electroanalysis and digital simulation. Jörg Strutwolf received the Diploma and Ph.D. degrees in the Theoretical Chemistry Group, University of Bielefeld, Germany. He has specialized in the investigation of interfacial transport processes by theoretical and experimental methods. His current research interests include the dynamics and reactivity of soft interfaces, the combination of microfluidics and electrochemistry, numerical modelling of transport and reaction phenomena in electrochemistry (mainly in co-operation with Dieter Britz), electrochemistry at the nanoscale, and nanostructuring of interfaces for sensor application. Currently, he is a Visiting Scientist at the University of Tübingen, Germany. He has worked in numerous electrochemistry groups, e.g. at University College London, U.K., University of Warwick, Coventry, U.K., Universitat Rovira i Virgili, Tarragona, Spain, and Tyndall National Institute, Cork, Ireland.

Preface 8
References 9
Contents 10
1 Introduction 17
References 20
2 Basic Equations 21
2.1 General 21
2.2 Some Mathematics: Transport Equations 22
2.2.1 Diffusion 22
2.2.2 Diffusion Current 24
2.2.3 Convection 24
2.2.4 Migration 25
2.2.5 Total Transport Equation 26
2.2.6 Homogeneous Kinetics 26
2.2.7 Heterogeneous Kinetics 28
2.3 Normalisation: Making the Variables Dimensionless 29
2.4 Some Model Systems and Their Normalisations 31
2.4.1 Potential Steps 31
2.4.1.1 Cottrell System 32
2.4.1.2 Potential Step, Reversible System 35
2.4.1.3 Potential Step, Quasi- and Irreversible System 37
2.4.1.4 Potential Step, Homogeneous Chemical Reactions 38
2.4.2 Constant Current 42
2.4.3 Linear Sweep Voltammetry 43
2.5 Adsorption Kinetics 47
References 51
3 Approximations to Derivatives 54
3.1 Approximation Order 54
3.2 Two-Point First Derivative Approximations 55
3.3 Multi-Point First Derivative Approximations 57
3.4 The Current Approximation 60
3.5 The Current Approximation Function G 60
3.5.1 Unequal Intervals 61
3.6 High-Order Compact (Hermitian) Current Approximation 61
3.7 Second Derivative Approximations 65
3.8 Derivatives on Unevenly Spaced Points 66
3.8.1 Error Orders 69
3.8.2 A Special Case 70
3.8.3 Current Approximation 70
3.8.4 An Example 71
3.9 The Fornberg Algorithm 72
References 73
4 Ordinary Differential Equations 75
4.1 An Example ode 76
4.2 Local and Global Errors 76
4.3 What Distinguishes the Methods 76
4.4 Euler Method 77
4.5 Runge–Kutta (RK) 78
4.6 Backwards Implicit (BI) 80
4.7 Trapezium Method 81
4.8 Backward Differentiation Formula (BDF) 82
4.8.1 Starting BDF 83
4.8.1.1 Time Shifts 84
4.8.1.2 Testing the Starting Protocols 85
4.9 Extrapolation 86
4.10 Kimble and White (KW) 87
4.10.1 Using KW as a Start for BDF 90
4.11 Systems of odes 91
4.12 Rosenbrock Methods 94
4.12.1 Application to a Simple Example ODE 97
4.12.2 Error Estimates 97
4.13 Padé Approximants 98
References 99
5 The Explicit Method 102
5.1 The Discretisation 102
5.2 Practicalities 103
5.3 Chronoamperometry and -Potentiometry 105
5.4 Homogeneous Chemical Reactions (hcr) 106
5.4.1 The Reaction Layer 108
5.5 Linear Sweep Voltammetry 109
5.5.1 Boundary Condition Handling 111
References 112
6 Boundary Conditions 114
6.1 Classification of Boundary Conditions 114
6.2 Single Species: The u-v Device 115
6.2.1 Dirichlet Condition 115
6.2.2 Derivative Boundary Conditions 115
6.3 Two Species 119
6.3.1 Two-Point Derivative Cases 123
6.4 Two Species with Coupled Reactions: U-V 124
6.5 Brute Force 130
6.6 A General Formalism 132
References 133
7 Unequal Intervals 135
7.1 Transformation 136
7.1.1 Discretising the Transformed Equation 138
7.1.2 Choice of Transformation Parameters 139
7.2 Direct Application of an Arbitrary Grid 140
7.2.1 Choice of Parameters 143
7.2.2 Current and C0 Approximations 144
7.3 Concluding Remarks on Unequal Spatial Intervals 144
7.4 Unequal Time Intervals 145
7.4.1 Implementation of Exponentially Increasing Time Intervals 146
7.5 Adaptive Interval Changes 147
7.5.1 Spatial Interval Adaptation 147
7.5.2 Time Interval Adaptation 151
References 152
8 The Commonly Used Implicit Methods 157
8.1 The Laasonen Method or BI 159
8.2 The Crank–Nicolson Method, CN 160
8.3 Solving the Implicit System 161
8.4 Using Four-Point Spatial Second Derivatives 163
8.5 Improvements on CN and Laasonen 166
8.5.1 Damping the CN Oscillations 168
8.5.1.1 First-Interval Subdivision 168
8.5.1.2 Initial BI Step(s) 169
8.5.1.3 Averaging and Extrapolation 170
8.5.1.4 Singularity Correction 171
8.5.1.5 Recommendations 171
8.5.2 Making Laasonen More Accurate 171
8.5.2.1 BDF 172
8.5.2.2 Extrapolation 174
8.6 Homogeneous Chemical Reactions 175
8.6.1 Nonlinear Equations 175
8.6.1.1 Linearising Squared Concentration Terms 176
8.6.1.2 Linearising the Product of Concentrations of Two Species 177
8.6.1.3 An Example Case: Linearising 177
8.6.1.4 An Example Case: Nonlinear 179
8.6.2 Coupled Equations 182
References 184
9 Other Methods 189
9.1 The Box Method 189
9.2 Improvements on Standard Methods 193
9.2.1 The Kimble and White Method 193
9.2.2 Multi-Point Second Spatial Derivatives 195
9.2.3 DuFort–Frankel 197
9.2.4 Saul'yev 198
9.2.5 Hopscotch 201
9.2.6 Runge–Kutta 203
9.2.7 Hermitian Methods 204
9.2.7.1 Numerov/Douglas 204
9.2.7.2 Hermitian Current Approximation 207
9.2.7.3 Method of Wu and White 209
9.3 MOL and DAE 210
9.4 The Rosenbrock Method 212
9.4.1 An Example, the Birk–Perone System 215
9.5 FEM, BEM, FVM and FAM (Briefly) 218
9.6 Orthogonal Collocation (OC) 219
9.6.1 Current Calculation with OC 226
9.6.2 A Numerical Example 226
9.7 Eigenvalue–Eigenvector Method 228
9.8 Integral Equation Method 231
9.9 The Network Method 232
9.10 Treanor Method 233
9.11 Monte Carlo Method 233
References 234
10 Adsorption 247
10.1 Transport and Isotherm Limited Adsorption 248
10.2 Adsorption Rate Limited Adsorption 250
References 250
11 Effects Due to Uncompensated Resistance and Capacitance 253
11.1 Boundary Conditions 255
11.1.1 An Example 257
References 260
12 Two (and Three) Dimensions 262
12.1 Theories 263
12.1.1 The Ultramicrodisk Electrode 263
12.1.1.1 Ranges of Applicability 269
12.1.1.2 LSV 270
12.1.2 Other UMEs 271
12.1.3 Some Relations 273
12.2 Simulations 274
12.3 Simulating the UMDE 276
12.3.1 Methods of Solution 277
12.3.1.1 Hopscotch 277
12.3.1.2 ADI 277
12.3.1.3 Some Other Methods 278
12.3.2 Direct Discretisation 279
12.3.3 Discretisation in the Mapped Space 287
12.3.3.1 Some Transformations 288
12.3.3.2 Inversion of the Transformations 292
12.3.3.3 The Diffusion Equation in the Mapped Spaces 294
12.3.3.4 Current Integration in Conformal Coordinates 295
12.3.4 Band Electrodes 296
12.3.4.1 Choice of ?max 297
12.3.4.2 Discretisation 297
12.3.4.3 Unequal Intervals in the Mapped Space 299
12.3.5 A Remark on the Boundary Conditions 299
12.4 Three-Dimensional Simulations 300
12.4.1 Square and Rectangular UMEs 300
12.4.1.1 Discretisation 304
12.4.2 The Grid 307
12.5 Ultramicroelectrode Arrays 308
12.5.1 Regular Arrays of UMDEs 309
12.5.1.1 The Diffusion Domain Approach 311
12.5.1.2 An Example 313
12.5.2 Arrays of UMBEs 317
12.5.3 Elevated UMBEs 321
12.5.4 Dual Electrode Systems 325
References 330
13 Migrational Effects 349
13.1 Theory 350
13.2 Simulations 352
13.3 Time Development of a Liquid Junction 352
13.3.1 Normalisation 353
13.4 RPC Example 362
13.5 Copper Deposition on an RDE 368
13.5.1 Note on Normalisations 373
References 375
14 Convection 378
14.1 Some Fluid Dynamics 378
14.1.1 Layer Relations 382
14.2 Electrodes in Flow Systems 383
14.3 Simulations 384
14.4 A Simple Example: The Band Electrode in a Channel Flow 385
14.5 Normalisations 386
References 390
15 Performance 398
15.1 Convergence 398
15.2 Consistency 400
15.3 Stability 401
15.3.1 Heuristic Method 402
15.3.2 Von Neumann Stability Analysis 403
15.3.3 Matrix Stability Analysis 405
15.3.3.1 Using Eigenvalues 406
15.3.3.2 Using the Matrix Norm 411
15.3.4 Some Special Cases 412
15.4 The Stability Function 412
15.5 Accuracy Order 415
15.5.1 Order Determination 415
15.6 Sensitivity Analysis 418
15.7 Accuracy, Efficiency and Choice 418
15.7.1 Determining Accuracy 422
15.8 Two- (and Three-)Dimensional Problems 423
15.9 Summary of Methods 423
References 425
16 Programming 430
16.1 Language and Style 430
16.2 Debugging 431
16.3 Libraries 433
References 433
17 Simulation Packages 435
17.1 Kinetic Compilers 438
17.2 Parameter Estimation 439
References 441
Erratum to: Digital Simulation in Electrochemistry 447
A Tables and Formulae 449
A.1 First Derivative Approximations 449
A.2 Current Approximations 449
A.3 Second Derivative Approximations 449
A.4 Unequal Intervals 450
A.5 Jacobi Roots for Orthogonal Collocation 454
A.6 Rosenbrock Constants 455
References 456
B Transforming the Diffusion Equation into Curvilinear Coordinates 457
B.1 Introduction 457
B.2 A Simple Example: Cartesian to Cylindrical 458
B.3 Transformations for the Band Electrode 460
B.3.1 Cartesian to MWA 460
B.3.2 Extension to VB 460
B.4 Disk Electrode Transformations 461
B.5 Transforming the Current 462
References 463
C Some Mathematical Proofs 464
C.1 Consistency of the Sequential Method 464
C.2 The Feldberg Start for BDF 465
C.3 Similarity of the Exponential Expansion and Transformation Functions 471
References 473
D Finding ?max 474
References 476
E Procedure and Program Examples 477
E.1 Example Modules 477
E.1.1 Module STUFF 477
E.1.2 Module ROSTUFF 478
E.2 Procedures 479
E.2.1 File Names Routine 479
E.2.2 The Error Functions 480
E.2.3 Current Approximations 480
E.2.4 Matrix Inversion 480
E.2.5 MINMAX 481
E.2.6 EE_FAC 481
E.2.7 DAMPED_EXPANSION 481
E.2.8 SV_FAC 482
E.2.9 Gradient Routine FORNBERG and FORN 482
E.2.10 Current Integration on an Unequally Gridded Surface 482
E.2.11 Reference Fluxes and Errors 482
E.2.12 JCOBI 483
E.2.13 I1I2 483
E.3 Example Programs 483
E.3.1 Program COTT_EX 483
E.3.2 Program CHRONO_EX 484
E.3.3 Program CV_EX 484
E.3.4 Program COTT_CN 485
E.3.5 Program CHRONO_CN 486
E.3.6 Program CHRONO_CN_HERM 486
E.3.7 Program LSV_CN 487
E.3.8 Program COTT_EXTRAP 487
E.3.9 A Nonlinear System: Programs for the Birk/Perone Reaction 487
E.3.10 EC Reaction, Cyclic Voltammetry: CV_EC 488
E.3.11 CV of the EC' Reaction: Program CV_CAT 489
E.3.12 LSV Simulation with iR Drop and Capacitance: Program LSV4IRC 489
E.3.13 Program UMDE_DIRECT 490
E.3.14 Program UMDE_VB 490
E.3.15 Program UMDE_ARRAY 490
E.3.16 Programs LIQU_JUNC, RPC, CURDE 491
E.3.17 Program CHANNEL_BAND 491
References 491
Index 493